Stationary BTZ space-time in Ricci-Inverse and $f(\mathcal{R})$ gravity theories

Abstract

In this paper, we explore a stationary BTZ space-time within the framework of modified gravity theory, specifically focusing on Ricci-inverse gravity. It is important to clarify that ``Ricci-inverse" refers to the inverse of the Ricci tensor, not the Ricci scalar. We consider a general class of this gravity theory, where the function $f$ is defined by $f(\mathcal{R},\mathcal{A},A^{\mu \nu }{A}_{\mu \nu })$, with $\mathcal{R}$ and $\mathcal{A}$ representing the Ricci and anti-curvature scalars, respectively and $A^{\mu \nu }$ is the anti-curvature tensor. We demonstrate that stationary BTZ space-time is a valid solution in this gravity theory, wherein the cosmological constant undergoes modifications due to the coupling constants. Moreover, we study another modified gravity theory known as $f(\mathcal{R})$-gravity and analyze the stationary BTZ space-time. Subsequently, we fully integrate the geodesic equations for BTZ space-time constructed within the Ricci-Inverse gravity, expressing the solutions in terms of elementary functions and compared with the GR result. We classify different types of geodesics, including null and time-like geodesics, under three conditions: (i) nonzero mass and angular momentum, $M\ne 0,J\ne 0$, (ii) massless BTZ space-time, $M=0$ and $J=0$, and (iii) $M=-1,J=0$, that is $Ad{S}_3$-type, and analyze the results in modified gravity theories and compare with the general relativity case.

31 pages, 12 figures, published in EPJC (https://doi.org/10.1140/epjc/s10052-024-13637-1)

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